数列极限42 课件

1、2019/3/19,微积分,数列极限,1,上,课,2019/3/19,微积分,数列极限,2,第,2,章,极限与连续,2019/3/19,微积分,数列极限,3,一、数列概念,数列,可看作自变量为正整数的,函数,下标函数,n,y,u,f,n,2.1,数列的极限,2,特性,1,有界性,2,单调性,1,定义,按自然数编号依次排列的一列数,1,2,n,u,u,u,称为无穷数列,简称数列,记为,u,n,其中的每个数,称为数列的项,u,n,称为通项,一般项,数列,u,n,若,正数,M,使得一切自然数,n,恒,有,称数列,u,n,有界,否则,称为无界,n,u,M,1,2,n,u,u,u,称此数列单调增加,1,

2、2,n,u,u,u,称此数列单调减少,2019/3/19,微积分,数列极限,4,割之弥细，所,失弥少，割之又,割，以至于不可,割，则与圆周合,体而无所失矣,1,早期极限思想的体现,放映,1,二、数列极限概念,当自变量,n,趋于无穷大时，数列,y,f,n,的变化趋势,y,n,1,刘徽的割圆术,极限：研究函数在自变量的某个变化过程中，函,数值无限趋近于某个常数的性质,对于数列,2019/3/19,微积分,数列极限,5,R,正六边形的面积,1,A,正十二边形的面积,2,A,正,形的面积,1,2,6,n,n,A,3,2,1,n,A,A,A,A,S,2019/3/19,微积分,数列极限,6,2,庄子的截

3、丈问题,1,2,2,1,2,1,2,n,第一天剩余,u,1,第二天剩余,u,2,第,n,天剩余,u,n,1,2,n,0,但,0,n,时,u,n,一尺之棰,日取其半,万世不竭,1,1,1,1,2,4,8,1,6,1,0,1,8,1,2,1,n,2,1,0,2,1,n,n,时,当,3,1,2,4,2,3,4,5,2,0,1,2,1,3,2,4,3,1,n,n,1,1,n,n,n,时,当,3,1,1,1,1,0,1,1,间,摆,与,在,时,当,1,1,1,1,n,n,1,4,l,i,m,n,n,n,ua,u,a,n,或,4,3,6,1,2,2,4,2,直观定义,数列,u,n,若当,n,无限增大时,u

4、,n,无限趋,近于常数,a,则称数列,u,n,以,a,为极限,或称,u,n,收敛,于,a,记,1,2,n,n,u,1,n,n,n,u,1,1,n,n,u,1,3,2,n,n,u,1,3,2,n,n,发,散,无限增大,例,否则称,u,n,发散,2019/3/19,微积分,数列极限,8,1,1,1,时的变,当,观察数列,n,n,n,播放,对于较简单的数列的极限,可通过观察法求得，例,1,1,5,1,1,l,i,m,l,i,m,2,l,i,m,l,n,1,l,i,m,l,i,m,1,n,n,n,n,n,n,n,n,n,e,n,n,n,n,n,0,2,0,1,1,lim,1,n,n,n,0,数列,极限

5、,的,严格,定义,2019/3/19,微积分,数列极限,9,1,1,1,1,n,n,n,u,n,当,无,限,增,大,时,无,限,接,近,于,问题,无限接近”意味着什么,如何用数学语言刻划,它,1,n,u,n,n,n,1,1,1,1,100,1,给定,100,1,1,n,由,100,时,只要,n,1,1,1,0,0,n,u,有,1000,1,给定,1000,时,只要,n,1,1,1,0,0,0,0,n,u,有,10000,1,给定,10000,时,只要,n,1,1,1,0,0,0,n,u,有,0,给定,1,时,只要,N,n,1,n,u,有,成,立,10,3.,N,定义,例,1,1,1,lim,1

6、,n,n,n,n,证明,证,1,1,1,n,n,n,n,1,0,1,n,u,要,1,n,只,要,1,N,取,1,1,1,n,n,n,有,1,1,l,i,m,1,n,n,n,n,故,l,i,m,0,d,n,n,n,ua,N,n,N,u,a,l,i,m,n,n,n,ua,u,a,n,或,设有数列,u,n,若对任意,总,则称,a,是数列,u,n,的极限，或称,u,n,收敛于,a,记作,0,存在正整数,N,使得当,n,N,时，恒有,n,u,a,成立,否则称数列,u,n,发散,则当,n,N,时,11,注,3,N,一般与任意给定的正数,有关,越小,N,越大,例,2,l,i,m,n,n,n,u,C,C,u,

7、C,设,为,常,数,证,明,证,n,u,C,C,C,成立,0,任给,0,n,对于一切自然数,l,i,m,n,n,u,C,故,说明,常数列的极限等于同一常数,n,u,a,1,具有二重性,任意性和不变性。在取,时,对其大小,不加限制，正由于这种,任意,性，才能用,刻划,u,n,与,a,任意接近。而在根据,找,N,时它是,不变,的,2,刻划,u,n,与,a,接近的程度,N,刻划数列作为动点运动到什,么时刻可使,u,n,与,a,接近程度小于给定的,若把数列看成函,数,则,N,分别用来刻划因变量及自变量的变化过程,4,N,是不唯一的，用定义证明数列极限时,关键是对任意,给定的,0,由,来,寻找,N,但不

8、必要求最小的,N,n,u,a,例,3,1,lim,0,2,n,n,证,明,证,1,0,2,n,1,2,n,0,0,n,u,要,2,1,log,n,只,要,2,1,log,N,取,1,0,2,n,有,1,lim,0,2,n,n,故,1,2,n,即,1,1,N,1,n,不妨设,1,则当,n,N,时,1,0,2,n,1,2,n,0,n,u,要,例,3,可用,放大手法,1,n,只,要,1,N,取,P31,例,2,取,要,设,1,注,1,放大”是为方便解不等式。注意不能,放过头,上,例,若将,放大为,1,则,1,不可能小于任意给定的正数,1,2,n,2,放大”后找到的,N,通常比不放大解得,若易解,的要

9、大,2019/3/19,微积分,数列极限,13,x,1,u,2,u,1,N,u,3,u,2,a,a,a,2,N,u,三、数列极限的几何意义,l,i,m,0,n,n,ua,N,则,正,整,数,使得数列从,a,a,a,的,邻,域,内,第,N,1,项起，以后所有项,u,n,1,u,n,2,都落在,至多只有,N,项落在该,邻域之外,l,i,m,0,d,n,n,n,ua,N,n,N,u,a,n,a,u,a,2019/3/19,微积分,数列极限,14,1,唯一性,定理,每个收敛的数列只有一个极限,证,l,i,m,l,i,m,n,n,n,n,u,a,u,b,设,又,由定义,1,2,0,N,N,使,得,1,n

10、,nN,u,a,当,时,恒,有,2,n,nN,u,b,当,时,恒,有,max,2,1,N,N,N,取,时有,则当,N,n,n,n,ab,u,b,u,a,n,n,u,b,u,a,2,时才能成立,上式仅当,b,a,故收敛数列极限唯一,四、数列极限的性质,2019/3/19,微积分,数列极限,15,收敛数列的有界性的图示,定理,收敛的数列必定有界,2,有界性,例如,1,n,n,u,n,数,列,2,n,n,u,数,列,有界,无界,数轴上有界数列的点,u,n,都落在闭区间,M,M,上,2019/3/19,微积分,数列极限,16,证,l,i,m,n,n,u,a,设,由定义,1,取,1,n,N,n,N,u,

11、a,则,使,得,当,时,恒,有,1,1,n,a,u,a,即,有,1,m,a,x,1,1,N,M,u,u,a,a,记,n,n,u,M,则,对,一,切,自,然,数,皆,有,n,u,故,有,界,注意,有界性是数列收敛的必要条件,推论,无界数列必定发散,定理,收敛的数列必定有界,l,i,m,n,n,u,a,设,l,i,m,0,d,n,n,n,ua,N,n,N,u,a,n,a,u,a,取,1,则,n,N,时,u,n,有界,2019/3/19,微积分,数列极限,17,五,小结,数列,研究其变化规律,数列极限,极限思想,精确定义,几何意义,收敛数列的性质,唯一性、有界性,思考题,1,试判断下列论断是否正确,1,若,n,越大,越接近于零,则有,n,u,a,lim,n,n,u,a,3,若对,存在自然数,N,当,n,N,时,数列,u,n,中,有无穷多项满足不等式,则有,0,n,u,a,lim,n,n,u,a,2,若,则,n,越大,越接近于零,n,u,a,lim,n,n,u,a,4,若对,数列,u,n,中除了有限项外都满足不,等式,则有,0,n,u,a,lim,n,n,u,a,3,从几何直观层次思考：若数列为单调